# Why does radioactive material decay exponentially as opposed to at a constant rate?

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Can anyone explain to me why exactly it is that radioactive material decays at an exponential rate? To make my confusion more clear, take the following example. Let's say you have 1 gram of radioactive material, and for the pure sake of argument, though I admit that I'm just pulling this number out of a hat, let's assume that 10 trillion radioactive atoms are in this gram. The half-life of this fictional radioactive material is 1 million years.

Now before I go any further, I just want to make it clear that I more or less understand the mathematics behind radioactive decay (N=N0 x e^-kt), so I don't need an explanation of that, or of how to use said formula.

Now, even though I know how to use that formula to calculate the amount left after n amount of time, I have always been completely baffled by the idea of it decaying exponentially. In other words, if you were to ask someone on the street who knew absolutely nothing about radioactive decay: "If you start with 1 gram of radioactive material, and after 1 million years it will have decayed to 0.5 grams, how long will it take for it to completely disappear?" I can almost guarantee that their answer will be 2 million years (that's what common sense would dictate), and for the life of me I have never been able to figure out why this would not be the case, because it is completely counter intuitive. Can someone explain to me what the mechanism is that makes radioactive decay an exponential process, as opposed to one that occurs at a constant rate? Do we even know?

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For example, suppose a particular isotope has a 50% chance of decaying during one year. If you have ten trillion atoms of this isotope in a sample, you'd expect about half to decay in a year. After another year, you'd expect half of the remainder to decay, and so on. Exponential decay.

You may wonder why the odds of an atom decaying don't increase over time. You might expect a ten-year-old atom to be more to decay than a brand-new atom. This could give you a constant decay rate, but it implies that atoms somehow "know" when to decay and decay faster the older they are. This isn't the case, just like how a losing streak of ten games in a row doesn't make you more likely to win the next game. Radioactive decay is stochastic -- atoms have no sense of time, and their odds of decaying are constant no matter their age.

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Think of atoms like coins. Heads is decayed. Tails is undecayed.

You have 100 of these coin atoms. Every ten minutes, you flip all of the coins and remove any that come up heads (decayed).

The first coin flip, you expect about half to come up heads, so you have fifty left. The second flip, you have half of those come up heads, leaving you with 25, and so on.

Because decay is a binary position, either the atom is decayed or it is not in the same way that a coin is either heads or tails but never 1/3 tails and 2/3 heads. There is no process of decaying, merely a probability of decay occurring at any given moment.

Or thinking of it another way, the atoms don't "know" how big of a chunk of material it is in, and that will have no impact on when the atom decays. An atom in one pound of a substance will not decay any sooner or later than an atom in a ton of the same substance.

So, if you start with a gram and a half a gram of atoms. After the half life has elapsed, you will have a half of a gram and a quarter of a gram. The half a gram doesn't know it started out as a gram, so it's like starting that exact moment with half a gram, which will take just as long to halve itself as the original half a gram did.

This only seems weird to us intuitively because we are used to dealing with more mechanical processes that have lots of in between steps. For instance, if it takes me half an hour to ear half a tub of ice cream, then, barring changes like brain freeze or feeling sick, it should take me an hour to eat the entire thing.

However, there is not a set probability that any random bit of ice cream will be eaten at any given moment. I am far more likely to eat a bit of ice cream that is exposed on the surface than one buried at the bottom of the tub. Additionally, every bit I eat brings bits of ice cream farther down closer to the surface by lower where that surface is, this changes their probability of being eaten over time.

With decay, it is not being induced by something else (usually), the placement of the atom in the material does not impact the likelihood of the atom decaying and other atoms decaying elsewhere in the substance do not change the probability of the atom decaying.

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It's the fact you're measuring half life that makes it exponential, if you were to measure the number of decays per thousand atoms you would find it is constant regardless of how much you start with i.e. linear

Or to put it another way, 1/2 of 1/2 of 1/2 of 1/2 of... etc, is exponential

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I think that it is because of the fact that decay itself is based on probability, that cause it to be goin at a exponential rate.

For example, the half life of 1000 isotope is 10seconds.

So every 10s, the probability will kick in. Ideally, since the probability of decay is 1/2, 1/2 of the original number of isotope will remain, that is, 500.

As the next 10seconds is over, the probability will kick in again.

So this will make decay going at a exponential rate.

I personally thinks that why people in general misunderstood half life as the fact that if it takes 10seconds for half the isotope to decay, it will take another 10seconds for the rest of the isotope to decay, is because they view it as it takes 10seconds for 500 isotopes to decay. Therefore it will take another 10seconds for the rest to decay. They are not linear. Decay is a probability as I said above.

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• 3 weeks later...

Think of atoms like coins. Heads is decayed. Tails is undecayed.

You have 100 of these coin atoms. Every ten minutes, you flip all of the coins and remove any that come up heads (decayed).

The first coin flip, you expect about half to come up heads, so you have fifty left. The second flip, you have half of those come up heads, leaving you with 25, and so on.

Delta, many thanks for a thought-provoking post. But - isn't there an obvious objection to your coin/atom analogy:

A coin lies where it is, inert. Until some person picks it up - and gives it a "flip".

Who or what flips the atom?

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Delta, many thanks for a thought-provoking post. But - isn't there an obvious objection to your coin/atom analogy:

A coin lies where it is, inert. Until some person picks it up - and gives it a "flip".

Who or what flips the atom?

You're taking the analogy too far. The coin flip and the atom's decay are probabilistic events. The atom decays because it can — an unstable atom is not in its lowest energy state.

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Can anyone explain to me why exactly it is that radioactive material decays at an exponential rate? To make my confusion more clear, take the following example. Let's say you have 1 gram of radioactive material, and for the pure sake of argument, though I admit that I'm just pulling this number out of a hat, let's assume that 10 trillion radioactive atoms are in this gram. The half-life of this fictional radioactive material is 1 million years.

Now before I go any further, I just want to make it clear that I more or less understand the mathematics behind radioactive decay (N=N0 x e^-kt), so I don't need an explanation of that, or of how to use said formula.

Now, even though I know how to use that formula to calculate the amount left after n amount of time, I have always been completely baffled by the idea of it decaying exponentially. In other words, if you were to ask someone on the street who knew absolutely nothing about radioactive decay: "If you start with 1 gram of radioactive material, and after 1 million years it will have decayed to 0.5 grams, how long will it take for it to completely disappear?" I can almost guarantee that their answer will be 2 million years (that's what common sense would dictate), and for the life of me I have never been able to figure out why this would not be the case, because it is completely counter intuitive. Can someone explain to me what the mechanism is that makes radioactive decay an exponential process, as opposed to one that occurs at a constant rate? Do we even know?

Radioactive material cannot decays at a constant rate because it would imply radioactivity even when the material exhausts. Macroscopically you can measure the rate and check that is exponential. If you are looking for an explanation, then you must study the microscopic details. A microscopic study shows that the decay follows a first order kinetics

$\frac{dN}{dt}= -kN$

integrating this kinetic equation gives the exponential form of above.

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